Discrete-time system properties — match each input–output relation to linearity/causality List I (I/P → O/P) A. y(n) = x(n) B. y(n) = x(n^2) C. y(n) = [x(-n)]^2 D. y(n) = [x(n)]^2 List II (System property) 1. Nonlinear, non-causal 2. Linear, non-causal 3. Linear, causal 4. Nonlinear, causal

Introduction / Context:
Determining linearity and causality from an input–output equation is a core DSP skill. Linearity means superposition holds; causality means the output at index n depends only on inputs at indices ≤ n.

Given Data / Assumptions:

• x(n) is a generic discrete-time sequence.
• Time index n ranges over all integers.
• Standard definitions of linearity and causality apply.

Concept / Approach:
Check linearity via additivity and homogeneity; check causality by whether the formula references future samples (k > n) when computing y(n). Time reversal x(-n) uses future values for n > 0; time warping x(n^2) references x at indices larger than n for n > 1, breaking causality (though linearity may still hold if no nonlinear functions like squaring are present). Squaring makes a system nonlinear.

Step-by-Step Solution:

Verification / Alternative check:
Test superposition: for C and D, [ax1 + bx2]^2 ≠ a[x1]^2 + b[x2]^2, failing linearity. For causality: choose an impulse at a future index and observe whether current y(n) changes.

Why Other Options Are Wrong:

• Labeling B “causal” ignores that x(n^2) references future indices.
• Calling C “causal” or “linear” contradicts squaring and time reversal.

Common Pitfalls:
Confusing linear time-warps with linear systems; linearity refers to signals, not just index algebra.

Final Answer:
A-3, B-2, C-1, D-4